# Reactant conversion in homogeneous turbulence: Mathematical modeling, computational validations, and practical applications

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## Abstract

Closed form analytical expressions are obtained for predicting the limiting rate of mean reactant conversion in homogeneous turbulent flows under the influence of a binary reaction of the type *F*+*rO*→(1+*r*) *Product*. These relations are obtained by means of a single-point Probability Density Function (PDF) method based on the *Amplitude Mapping Closure* (Kraichnan, 1989; Chen *et al.*, 1989; Pope, 1991). It is demonstrated that with this model, the maximum rate of the mean reactants' decay can be conveniently expressed in terms of definite integrals of the parabolic cylinder functions. For the cases with complete initial segregation, it is shown that the results agree very closely with those predicted by employing a beta density of the first kind for an appropriately defined Shvab-Zeldovich scalar variable. With this assumption, the final results can also be expressed in terms of closed form analytical expressions which are based on the incomplete beta functions. With both models, the dependence of the results on the stoichiometric coefficient and the equivalence ratio can be expressed in an explicit manner. For a stoichiometric mixture the analytical results simplify significantly. In the mapping closure these results are expressed in terms of simple trigonometric functions. For the beta density model they are in the form of gamma functions. In all the cases considered, the results are shown to agree well with data generated by Direct Numerical Simulations (DNS). Due to the simplicity of these expressions and because of nice mathematical features of the parabolic cylinder and the incomplete beta functions, these models are recommended for estimating the limiting rate of mean reactant conversion in homogeneous reacting flows. These results also provide a valuable tool in assessing the extent of validity of turbulence closures for the modeling of unpremixed reacting flows. Some discussions are provided on the extension of the models for teating more complicated reacting systems, including realistic kinetics schemes and multiscalar mixing with finite rate chemical reactions in more complex configurations.

## Keywords

Probability Density Function Direct Numerical Simulation Equivalence Ratio Reactant Conversion Mapping Closure## Preview

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